Low-temperature specific heat has been measured and extensively analyzed on a series of La2-xSrxCuO4 single crystals from underdoped to overdoped regime. From these data the quasiparticle density of state in the mixed state is derived and compared to the predicted scaling law C-vol/TrootH=f(T/rootH) of d-wave superconductivity. It is found that the scaling law can be nicely followed by the optimally doped sample (x = 0.15) in quite a wide region of (T/rootH less than or equal to 8K/rootT). However, the region for this scaling becomes smaller and smaller toward more underdoped region: a clear trend can be seen for samples from x=0.15 to 0.069. Therefore, generally speaking, the scaling quality becomes worse on the underdoped samples in terms of scalable region of T/rootH. This feature in the underdoped region is explained as due to the low-energy excitations from a second order ( for example, antiferromagnetic correlation, d-density wave, spin-density wave, or charge-density wave order) that may coexist or compete with superconductivity. Surprisingly, deviations from the d-wave scaling law have also been found for the overdoped sample (x=0.22), while the scaling law is reconciled for the overdoped sample, when the core size effect is taken into account. An important discovery of present work is that the zero-temperature data follow the Volovik's relation Deltagamma(T=0)=ArootH quite well for all samples investigated here; although the applicability of the d-wave scaling law to the data at finite temperatures varies with doped-hole concentration. We also present the doping dependence of some parameters, such as the residual linear term gamma(0), the alpha value, etc. It is suggested that the residual linear term (gamma(0)T) of the electronic specific heat observed in all cuprate superconductors is probably due to the inhomogeneity, either chemical or electronic in origin. The field-induced reduction of the specific heat in the mixed state is also reported. Finally, implications on the electronic phase diagram are suggested.